So much misinformation about networks, and how they work. There are almost no differences in Cat cables, the difference is the bandwith that they work on, the higher the Cat rating, usually the higher quality of cable. However, there are good cables and bad cables of every category. For streaming, a simple C5 cable is fine. A certified cable are the ones to get, these are tested for their cat ratings. Most of my cables come from years of working in/around data centers. But Blue Jeans are great cables you can trust.
Streaming 16/44 takes around 5mb of bandwith, with all the overhead. A simple network can handle it, and then some. LTE phone data can do it. Usually on normal networks there is around 5mb of general overheard, like DNS, DHCP, handshakes, routing tables, etc...
There is no noise on ethernet, it's a packet transfer in the digital domain, there is no place for noise. If noise is coming through your cat cables you have crap equiptment, or another issue. There is noise on cheap wall warts, and of course WiFi is all noise.
Quality clocks on routers is not thing, they do not have the same jitter issues as a DAC, it's not as important. Never have I ever seen a clock rating on any networking device. It's not a thing that comes up on any network build out.
ISP supplied routers/WiFi are garbage! If you can get your own, do it, this is money well spent. But you will need some basic networking knowledge to set it up properly.
The only way to get a "clean" network is to have a firewall, or manage switch where ports an services are turned off, only allowing the traffic needed for streaming, blocking all the rest. Would not advice this unless you are a network engineer.
No networking gear should be in your audio rack, unless it's segregated from the rest, around 1m should be fine. RF/EMI from a router (no WiFi) will be around the same as your DAC. If it's a quality router, it should be less. A $50 router in a plastic case will be bad, a $300 router in a metal box (ie shielded) will be better.
There is no need to worry about cable length until it gets over 200ft. There is no packet degradation under that. Over that, fiber should be used. If you can swing it, a all fiber network is "best" but also most expensive.